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Full text of "An integrated-circuit piano tuner for the equal-tempered keyboard employing a tuneable fixed-coefficient digital filter [electronic resource]"

United States 
Naval Postgraduate School 




THESIS 



AN INTEGRATED-CIRCUIT PIANO TUNER 
FOR 
THE EQUAL-TEMPERED KEYBOARD EMPLOYING 

A 
TUNEABLE FIXED-COEFFICIENT DIGITAL FILTER 



by 



Michael William Hagee 



T/3Z&6S~ 



June 1969 



Tki& document na& been ajpptwvtd ^on. ptibLLc kz.- 
Iza&z and taJLz; itb dutsiibution u> tmtimitzd. 



An Inteqrated-Circuit Piano Tuner 

for 

the Equal -Tempered Keyboard Emoloyina 

a 

Tuneable Fixed-Coefficient Digital Filter 



by 



Michael Will iam i^laqee 
Second Lieutenant, United States Marine Corps 
B.S., United States Naval Academy, 1968 



Submitted in partial fulfillment of the 
requirements for the deqree of 



MASTER OF SCIENCE IN ELECTRICAL ENGINEERING 



from the 

NAVAL POSTGRADUATE SCHOOL 
June 1969 



ABSTRACT 

A study of the physics of the piano reveals that while the upper 
oartials of the steel strings are the eigen-frequencies of the complex 
tone, they are not integer multiDles of the resDective fundamentals. 
To nroDerly measure and tune these eigen-partials, a diqital filter 
caDable of sweeping a major portion of the audio-frequency spectrum 
had to be implemented. Such a filter, a tuneable fixed-coefficient 
digital filter, is discussed as well as a simple pole-zero design 
procedure for determining the required coefficients. Each module, 
including the Frequency Deviation Detector and Counter, the Time-Base 
Generator, the Digital Filter, the Reference Frequency Generator and 
the Display and Control Module, of the proposed tuner is illustrated 
and discussed. 



TABLE OF CONTENTS 

I. INTRODUCTION — 9 

A. SOUND AND PHYSICS OF THE PIANO 9 

B. MUSIC THEORY - 13 

C. STATEMENT OF THE PROBLEM — 17 

1. A Commercial Analog Device 17 

2. The Author's Proposed Imolementation 18 

II. DIGITAL FILTERING — - 19 

A. THE Z-TRANSFORM - - 20 

B. CALCULATION OF THE WEIGHTING COEFFICIENTS 21 

C. A TUNEABLE FILTER 23 

D. ERRORS THAT MUST BE CONSIDERED 28 

III. A PROPOSED DESIGN FOR A DIGITAL PIANO TUNER — 38 

A. INTRODUCTION AND OVER-ALL VIEW OF THE MACHINE — 38 

B. THE POWER SUPPLY - 35 

C. THE ANALOG TO DIGITAL CONVERTER 35 

D. THE FREQUENCY DEVIATION DETECTOR AND COUNTER MODULE — 37 

E. THE REFERENCE FREQUENCY GENERATOR MODULE 38 

F. THE TIME-BASE GENERATOR MODULE 42 

G. THE TUNEABLE DIGITAL FILTER MODULE 43 

H. THE DISPLAY AND CONTROL MODULE 44 

IV. CONCLUSION 45 

APPENDIX A GRAPHS 46 

APPENDIX B FIGURES 52 

COMPUTER OUTPUT — 75 

COMPUTER PROGRAM - — 77 



BIBLIOGRAPHY --- 90 

INITIAL DISTRIBUTION LIST -- - 95 

FORM DD 1473 - 97 



LIST OF TABLES 



I. Pairs of Tones Producing Agreeable Auditory Effects 15 

II. Reference Freguency Output Data for the Upper Three 

Octaves - 40 



LIST OF ILLUSTRATIONS 

1. Illustratina the Full Ranqe of the Piano with the Herz 
of Each Note in Equal TemDerament Based on the Standard 

Pitch of A440 — 52 

2. Vibrations of an Ideal String 53 

3. Tynical Vibrations of a Musical Strinq 54 

4. Beats Produced by a Dissonance Interval and a 

Consonance Interval 15 

5. Inharmonicity of the Steel -Strinqed Piano 55 

6. Pictorial Representation of an m -order Linear 

Difference Equation 56 

7. A General Pole-Zero Plot in the Z-Domain for a 

Diqital Passband Filter 57 

8. Cascaded Diqital Filters with a Constant Bandwidth 58 

9. General Block Diagram of the Pronosed Diqital Piano Tuner - 59 

10. Schematic of the Proposed Power Supply 60 

11. The Ramp A/D Converter 61 

12. The Ramp A/D Converter Waveforms 62 

13. The Frequency Deviation Detector and Counter Module 63 

14. Loqic Diaqram of the Four-Staqe Rinq Counter 64 

15. Waveforms of the FDDCM when the Unknown Input is SharD 65 

16. Waveforms of the FDDCM when the Unknown Input is Flat 66 

17. Waveforms of the FDDCM when the Unknown Input is 

Perfectly Tuned 67 

18. Block Diagram of the Reference Frequency Generator Module - 68 

19. Block Diagram of the Time-Base Generator Diagram 69 

20. Divide-by-Ten Circuit --- - 70 

21. Logical Schmitt Trigger 71 

22. Diode-Array Building Block for the Fixed-Coefficient 

Digital Filter - - 72 

7 



23. Illustrating a Three-Chip Second-Order Recursive Serial 

Digital Filter — 73 

24. Block Diagram and Decoding Scheme for a "10" Counter 74 



I. INTRODUCTION 

The purpose and end result of this paper 1s to demonstrate how 
a digital piano tuner may be constructed. After only a small amount 
of investigation, however, it became readily apparent that a certain 
understanding of not only the basic physics of the piano, but also 
of music theory, was essential before a true definition of the 
problem could be established. 

It was relatively simple to find many books on music and sound 
that gave the precise frequencies for each note of the "Well -Temper- 
ed Piano" (See Fig. 1). Using these frequencies 1t proved no 

great task to design and build a digital circuit that would in fact 

2 

measure, with a very high degree of accuracy, these various notes. 

It was soon found, however, that if this was done, such tuning would 
produce complete aural chaos to both the trained and untrained ear 
alike. As asserted above, to fully appreciate this "discovery" 
a small amount of musical and sound theory must be presented. 

A. SOUND AND PHYSICS OF THE PIANO 

When any string is struck or plucked and then allowed to vibrate, 
it not only vibrates at its fundamental frequency but also at its 
various upper harmonics or partials [Refs. MF-3, MJ-10 and MO-14]. 



Bach established the scale of equal temperament as an accepted 
scale in the musical world. His pieces for piano were all written in 
this scale. For a discussion of the history and explanation of the 
scale itself, see Refs. MH-5, M)-14, and MW-24. 

2 

See section head "A Frequency Deviation Detector and Counter Module." 



The fundamental frequency is determined by the length of the wire or 
string being considered and at what place along its length it is 
struck [Ref. MJ-10]. The upper partial s occur because of a certain 
amount of stiffness in almost all strings, especially the steel 
strings used in the piano. 

Fig. 2 illustrates the vibration of an ideal string, that is, 
one without any stiffness. This string can be made to vibrate at 
many different frequencies. The fundamental frequency (a) produces 
a pure tone rarely heard in music. The higher-pitched partial tones, 
or overtones, are produced by harmonic vibrations (b) and (c), whose 
frequencies are integer multiples of the fundamental frequency. 

Fig. 3 illustrates the simultaneous vibration of a string at 
two or more different frequencies and illustrates the mode that is 
normal for all stringed instruments. 

The complex sound produced by this combination of separate tones 
has a timbre, or characteristic quality, that is determined largely 
by the number of partial tones and their relative loudness [Refs. MB-1 , 
MC-2, MJ-10, MM-12, MP-19 and MO- 14]. It is timbre that enables one 
to distinguish between two musical tones that have the same pitch 
and loudness, but are produced by two different musical instruments. 
With this in mind, it seems reasonable to conclude that the upper 
partials and not the fundamental frequency are the determining factors 
in the sound that is perceived [Ref. MC-2]. 

In order to further substantiate this conclusion, a test done by 
Bell Telephone at the Smithsonian Institute several years ago was 
investigated. The Bell engineers made two recordings of musical 



10 



instruments, opera singer, noises and everyday sounds. In one, the 
fundamental frequency was extracted, while the other was left untouched, 
When played before both laymen and professionals in different fields, 
the groups were unable to determine which recording had the missing 
fundamental! In fact, even when all the frequencies of a musical 
composition below 300 hertz were removed, the quality of the music 
remained the same to a surprising degree [Ref. MC-4]. The "case of 
the missing fundamental" has been attributed to the fact that the 
ear uses periodicity and not frequency as a basis for pitch percep- 
tion [Ref. MP- 18]. 

Another interesting fact of similar nature 1s brought out in 
Refs. MC-2, and MJ-10. They show, for the modern piano, that although 
the fundamental is the "loudest" when the key is first struck it 
rapidly dies out and upper partials, having a much longer decay time, 
take over the pitch of the note. (In very old pianos, the fundamen- 
is actually absent!) All of this further substantiates the conclu- 
sion above that the upper partials are the most important fre- 
quencies in a complex musical sound. 

"So what?," may very well be the question in the mind of the 
reader at this time. With a seemingly logical approach, it could 
very well be asserted that when the fundamental is measured, the 
upper partials are also measured and once again only a relatively 
simple frequency measuring scheme is required. This is, however, 
not the case. 

As stated before, piano strings are made of steel because of 
the tremendous stresses they are required to withstand in order to 



11 



be able to generate an appreciable amount of sound. This use of 
a "wire" or steel string, unlike the gut string of the violin, 
introduces a certain amount of stiffness or inelasticity. This 
inelasticity has the very undesirable effect on the upper parti als 
of causing them to vibrate at frequencies that are not integer 
multiples of the fundamental frequency . No longer can the formula: 

f n " " f o «) 

J.L. 

be applied in determining the n partial. The formula is shown 



in Ref. MF-3 to be: 



f n = nf o ((l + Bn 2 ) / (1 + B)) 1/2 (2) 



where B is dependent on the dimensions of the wire! B is in fact 
a function of the length squared, diameter, Young's modulus of 
elasticity, the area of the cross section, the radius of gyration 
squared and the tension of the wire when in the neutral position. 
As can be easily imagined this causes the upper partials to differ 

from piano to piano and makes completely useless any attempt to tune 

3 
by zero beating the fundamentals. It should be noted here that the 

inharmonicity caused by this string stiffness always makes the 

upper partials sharp with respect to the desired, pure, integer 

harmonic. It is less pronounced in the lower octaves and greater 

in the higher registers . 



3 
In Ref. MM-13, Professor Franklin Miller suggests a means 

whereby the desirable result of reducing this inharmonicity of 

partials to a negligible quantity by applying a small amount of 

mechanical loading near one end of the piano string. 



12 



With the above comments made, it appears appropriate to make a 
small digression into the field of music theory and see what effect 
this inharmonicity has on a musical score. 

B. MUSIC THEORY 

The average human ear can distinguish about 1,400 discrete 

frequencies. However, in the equally tempered scale covering the 

3 
hearing range from 16 to 16 x 10 hertz, there are only 120 dis- 

4 
crete tones. 

Comparatively few people are capable of recognizing the true 
pitch of a musical tone. A great many individuals are able, though, 
to distinguish the ratio of two frequencies. Furthermore, most 
people recognize that when two notes are sounded together or immedi- 
ately following one another they either produce a pleasing effect or 
give a decidedly unpleasant reaction. In music theory these reactions 
are termed consonance and dissonance, respectively. The frequency 
ratios that produce these conditions are termed musical intervals or 
simply intervals. 



4 

Although really beyond the scope of this paper, the equal tem- 
pered scale brought to the fore-front by Bach in his pieces for the 
"Well Tempered Klavier" is of great importance in music theory. The 
scale allows the musician or composer to modulate or change keys of a 
composition without changing the entire frequency spectrum of the 
piano. This latter change in the frequency spectrum for every chanqe 
in key was why the more pleasing-to-the-ear scale of just intonation 
was finally dropped. It is interesting to note that all music before 
Bach's time was written in the scale of just intonation and modern 
day audiences are therefore not hearing these pieces as composed 
because of the change in scale, For a much deeper and interesting 
discussion of this subject see Refs. MH-6, MJ-9, M0-14, MP-15, 
Mw-21 and MW-24. 



13 



The question now arises: why when two notes e.g., C4 (261.63 Hz)" 

and D4 (293.66 Hz) are sounded together a very rough sensation is 

produced, while when C4 and E4 (329.63 Hz) are played, a very smooth 

and pleasing musical effect occurs., The answer was found first by 

the great investigator Helmholtz: 

When two musical tones are sounded at the 
same time, their united sound is generally 
disturbed by the beats of the upper par- 
tfals (harmonics), so that a greater or 
less part of the whole mass of sound is 
broken up into pulses of tone, and the joint 
effect is rough. This relation is called 
dissonance. But there are certain deter- 
minate ratios between frequency numbers, 
for which this rule suffers an exception, 
and either no beats at all are formed, or 
at least only such as have so little inten- 
sity that they produce no unpleasant dis- 
turbance of the united sound. These 5 
exceptional cases are called consonances. 

Helmholtz also found that this "roughness" was maximized at 33 beats/ 

sec and went as high as 132 beats/sec before its effect on musical 

sound became unappreciable. 

Now returning to the original question, concerning the differ- 
ence between C4-D4 (a musical second) and C4-E4 (a musical third), 
let the whole notes in Fig. 4 represent the notes played and the 
stars their respective upper partial s. 

It becomes readily apparent from the illustration that the beating 
of upper parti als of the second produce a beat frequency well within 
Helmholtz's "critical area," whereas the musical third does not. 



5 
Helmholtz, H.L.F. , On the Sensations of Tone , 2d Eng. ed., 

trans. Ellis, A., p. 179^197, Dover Publications, 1954.' 



14 



§ * 
* * 


2349.3 
2093.0 
256.3 b/s 

1174.7 
1046.5 
128.2 b/s 

587.33 
523.25 




i * 

i ! 


2637.0 
2093.0 
544.0 b/s 

1318.5 
1046.5 
272.0 b/s 

659.26 


* 




_^ it 


523.25 


# * 


64.08 b/s 

293.66 
261.63 
32.03 b/s 

4. Beats produce 
and a consonance 


136.01 b/s 








329.63 
261.63 
68.00 b/s 

nterval (a) 






r^ 


6 O 
C4 D4 
(a) 

Fig. 


id by a d 
interval 


- 

C4 E4 
(b) 

issonance i 
(b). 





The intervals that produce pleasing sensations have long been 
known to musicians and composers. It is the mixinq of these various 
deqrees of smooth harmonious sounds by themselves and with some of the 
dissonance intervals that creates all of the music heard today. Table 1 
gives the various consonant intervals in order of decreasing agreeable 
auditory sensations. Notice the apoearance of the musical third, while 
the second is conspicuously absent. 





NAME 




RATIO 


NAME 


RATIO 


Unison 




1::1 


Minor Third 


6::5 


Octave 




2::1 


Major Sixth 


5::3 


Fifth 




3::2 


Minor Sixth 


8::5 


Fourth 




4::3 


Major Seventh 


15: :8 


Major Thi 


rd 


5::4 


Minor Seventh 


9::5 


Table 1. Pa 


irs 


of tones producing 


agreeable auditory 


effects. 





15 



Before attempting to tie all of this together, a recent investiga- 
tion by Plomb [Ref. MP-18] will lay the foundation for the binding. 
Plomb conducted this investigation to determine what factor or factors 
in a complex sound caused a listener to assign it a definite Ditch. 
The results are therefore not only applicable to the complex sounds 
generated by the piano but can be applied to any complex tone. Plomb 
found that for fundamental frequencies up to about 350 hertz, the oitch 
was determined by the fourth and higher partial s 5 for frequencies up to 
about 700 hertz, by the third and higher partials; and for frequencies 
up to about 1400 hertz, the complex tone was determined by the second 
and higher partials. In all of the above cases, the fundamental was 
shown to have no bearing on the determination of the pitch once the 
tone had been generated. For frequencies above 1400 hertz, however, 
the fundamental appeared to be the determining factor. This was 
attributed to the fact that the ear starts having trouble detecting 
the periodicity of the tone at these high frequences. 

It now takes little imagination to guess what would happen if the 
previously discussed inharmonicity of the piano wires were taken into 
account in the example of the second and third (see Fig. 4). The upper 
partials of D4 would need to be sharpened by only a very small amount 
before this interval would cross Helmholtz's critical area and become 
quite rough and unpleasant to the listener. 

Thus the question arises: Just how much does the stiffness of the 
string cause the upper harmonics to vary? This question is easily 



16 



answered graphically in Fig. 5 [Ref. MB-1]. As is readily apoarent, 
the higher partial s can be as much as two full semi -tones sharp. 



C. STATEMENT OF THE PROBLEM 

In the previous discussion it has been shown that while all musical 
strings vibrate at their fundamental frequency, they also vibrate at 
their various upper oartials or harmonics. It was also pointed out 
that the upper partial s and not the fundamental determines the pitch 
of the complex sound. This dependency on harmonics would present no 
real problem in the digital tuning of the piano if the strings of the 
piano vibrated at partials that were integer multiples of the fundamen- 
tal . This, however, was shown not to be the case. Therefore in order 
to properly tune the steel-stringed piano, one of the upper partials, 
as determined by Plomb, must be filtered from the comolex sound, meas- 
ured and precisely tuned as if it exhibited the characteristics of a 
pure harmonic. 

1 . A Commercial Analog Device 

Conn Instruments Inc. now has the only instrument on the market 
that attemots to solve this problem [Ref. MK-11]. It is an analog 
calculator and filter that will measure the upoer partials of the 
various keys of the piano. However, Conn admits that its tuner, "The 
Strobotuner," requires the aid of a good pair of musical ears. 



A semi -tone is the difference in frequency between two adjacent keys 
in the equal -tempered scale. This amounts to about a six per cent differ- 
ence in frequency between notes. Musicians call 1/100 of a semi-tone a 
cent. 

It is not desired to degrade from the quality of this instrument in 
any way. It is in fact a very fine machine that goes a long way in at- 
tempting to solve a very complex problem. Further information may be ob- 
tained by writing Conn Instruments Inc. Elkhart, Indiana 46514. 



17 



This author believes this is due to the fact that the bearing is laid by 
measuring the fundamentals of the octave chosen and then usinq nothing 

o 

higher than the first partial in all subsequent tuning. As Plomb's 
investigation has already shown, this is by no means sufficient for 
the majority of the notes on the piano. 

This writer decided that a better tuner could be built if one 
would filter, measure and tune the partial s that Plomb's investigation 
proved to be of prime importance. 

2. The Author's Proposed Implementation 

The original goal was the desiqn of a digital oiano tuner. 
This goal resulted in the decision that some sort of digital filter 
had to be implemented. Although there has been a great deal of work 
done in the field of digital filtering [Refs. EB-4, EC-7 thru EC-10, 
EG-15, EG-17 thru EK-19, EK-21 , EM-28, EN-30, ER-33, ER-37, ER-38, 
ET-42 thru EV-46, EW-48 and EW-49] , the area of tuneable fixed-coef- 
ficient digital filtering has been seemingly completely untouched. 

Since the filtering requirements presented by the piano 
necessitated a filter that could sweep the frequency spectrum of the 
piano, it was decided to investigate this area of tuneable digital 
filtering. Although the remainder of this oaoer deals with the design 
of the proposed tuner, the stress was laid on the theory and imnle- 
mentation of this filter. 



"Laying the bearing" refers to the tuning of one octave, usually 
the one below A4 (i.e., C3-C4), by measuring its fundamentals very 
precisely and then using their upper partial s for all subsequent 
tuning. This usually results in less than perfect results. See 
Refs. MH-5, MW-22 and MW-25. 



II. DIGITAL FILTERING 

Digital filtering is the process of spectrum shaping using items 
of digital hardware as the basic building blocks. Thus the aims of 
digital filtering are the same as those of continuous filtering, but 
the physical realization is different. Linear continuous filter theory 
is based on the mathematics of linear differential eguations; linear 
digital filter theory is based on the mathematics of linear difference 
eguations. 

+■ h 

An m order linear difference eguation may be written as 

r m 

y(nT) = i L.x(nT - iT) - z K.y(nT - iT) (3) 
i=0 1 i=l n 

This form emphasizes the iterative nature of the difference eguation; 
given the m previous values of the outDut y and the r + 1 most recent 

values of the input x, the new output may be computed from (3). 
Physically, the input numbers are samples of a continuous waveform 
and real-time digital filtering consists of performinq the iteration 
of (3) for each arrival of a new input sample. Design of the filter 
consists of finding the constants K. and L. to fulfill a given filter- 
ing requirement. Real-time filtering implies that the execution time 
of the "computer program" for computing the riqht side of (3) is 
less than T, the sampling interval. 

See Fig. 6 for a pictorial representation of (3) consisting of 
unit delays of time T, adders and multipliers. 

It will be assumed in all further calculations that the set of 
input and output samples x(nT) and y(nT) are zero for all values of 
n less than zero. 



19 



A. THE Z-TRANSFORM 

The z-transform of a sequence x(nT) is defined as 

00 

X(Z) = z x(nT)Z' n (4) 

n=0 

where Z" is the unit delay operator defined in the s domain as 

Z = e st (5) 

where s = a + jw. 

For many sequences, the infinite sum of (4) can be expressed in 

closed form. For example, the z-transform of the sequence 

x(nT) = for n<0 (6) 

x(nT) = 1 for n^O (7) 

is 

X(Z) = E Z" n = l/d-Z" 1 ) (8) 

n=0 

The transform variable Z is, in general, a complex variable and 

X(Z) is therefore a function of a complex variable. 

The transfer function of the filter may now be written as 

H(Z) = Z(output)/Z(input) (9) 

m r 

= E KZ" n / (1 + z LZ" n ) (10) 

n=0 n n=l n 

The coefficients of the Z" terms correspond to the value of the 

weighting sequence at t = nT, where n is an integer. This transfer 

function, in order to be physically realizable, must not contain 

any positive power in Z. A positive power would indicate a prediction 

or simply that the output signal precedes the input. This condition 



20 



implies that m>r. When m = r, as in this project, L must not be 
— o 

zero. In order to ensure this condition is met, L has been set equal 
to one in (10). 

As asserted before, the problem is now one of finding the proper 

coefficients L and K . 

m n 

B. CALCULATION OF THE WEIGHTING COEFFICIENTS 



In the z-domain the zeroes may be written as 
Z 
and the poles as 



Z Q = R Q e jVr (11) 



Z p = y> T (12) 

and the recursive filter transfer function may then be written as: 

(1-Z Z" 1 ) (1-Z z" 1 ) — (i-z Z" 1 ) 
H(Z) = SL— ^2 2L_. (13) 

(1-Z ,Z"') (1-Z Z"')-~ (1-Z Z' 1 ) 

v pi ' v p2 ' v pm ' 

As stated above both r and m have been set equal to n in order to 
meet the conditions for real izabil ity. 

There remains now the step of multiplying out (13) and matching 
the derived coefficients with those of (10). The desired coeffi- 
cients are: 

K l ■ Z ol + Z 02 + Z 03 + — + Z on < 14 > 

K. = sumi of the products of the Z 's (15) 
taken k at a time. 

K = Z ,Z Z ,— Z (16) 

n ol o2 o3 on v ' 



and 



H = Z pl + Z P 2 + Z P 3 + - Z pn < 17 ' 

L, = sum of the products of the Z 's (18) 
taken k at a time. p 

L = Z ,Z 9 Z - — Z „ (19) 

n ol o2 o3 on 



21 



Due to the binomial coefficient nature of these terms and due to 
the fact that all of the terms are complex variables, a Fortran pro- 
gram was written to determine the various L's and K's for n as high 
as 50. Both the program and the calculated coefficients may be found 
in the computer section of this work. 

The frequency response of the designed filter may be obtained 
from: 

m . T r . _ 

H(a)) = E K/ Jul / (1 + z L e" Jwl ) (20) 
n=0 n n=0 n 

This equation is also imDlemented in the program mentioned above. 

In selecting the required poles and zeroes, two apparent limi- 
tations had to be imposed: 

1. The poles and zeroes had to be real or complex conjugates 
in order that the desired coefficients would have real 
values. 

2. The poles had to lie within the unit circle to Droduce 
a stable filter. 

From the theory of sampled-data systems [Refs. EK-20, EM-27 and ER-34] 
it can be shown that the oole-zero configuration of Fig. 7, when imple- 
mented, results in a passband with a center frequency determined 
by 

f c = 9 C / (2-itT) (21) 

The first factor to be determined is the distance at which to 
place the poles and zeroes around the center angle 8 in order to 
obtain the desired response. A complete investigation of this 
problem was made by Mooney in Ref. EM-28; the results will only be 
briefly summarized here. 



22 



Mooney found that close spacing of the poles and zeroes produced 
a sharD attenuation outside of the passband. This may easily be visua- 
lized if one thinks back on the effect the placement of poles and 
zeroes in the s-domain has on the sharpness of a continuous filter. It 
was also shown by Mooney that the passband was relatively insensitive 
to the location of the zeroes between the origin and the pole. How- 
ever, pole placement was critical and for the proper response should 
be placed as illustrated in Fig. 7. For graphs and drawings of how 
different placement affects the response of the designed filter see 
the cited reference. 

C. A TUNEABLE FILTER 

This author reasoned that since 

e = ml (22) 

than it should immediately follow that 

e, = o)-|T (23) 

e 2 = oj 2 T (24) 

where w, and u« are the respective lower and upper cut-off freguencies 
of the passband. Subtracting (23) from (24) gave 

6~ - 8, = (u>2 - u'i )T (25) 

A9 = AcoT (26) 

Aw = A6/T (27) 

where Aw could now be defined as the bandwidth of the filter. 

It was then assumed that A6 could be fixed, that that the poles 

and zeroes could be "hard-wired" in place in the z-domain. 



23 



Equation (27) would then reduce to 

Aw = C/T (28) 



or 



Aw = C ] f s = BW (29) 

where f is the sampling frequency, C and C-, are constants determined 
by A6 and BW is the bandwidth of the filter. 

Following the same reasoning it was easily shown that 

e c = coj (30) 

e c ° = 360 f c / f s (31) 

where f is the center frequency of the filter and e the associated 

angle in the z-domain. Once again making the angles e, and e 2 
fixed in the z-domain, (31) could be written 

f c = C 2 /T = C 3 f s (32) 

Equations (29) and (32) are the ones of prime interest. Equation 
(32) illustrates that the center frequency f is directly related 
to the sampling frequency f . Therefore by changing f it should be 
able to cause the filter to sweep the entire frequency spectrum with 
a bandwidth determined by (29). However, (29) immediately illustrates 
the existence of the ever-present "trade-offs" encountered in design. 
That is as the filter is caused to sweep and tune to ever increasing 
values of the frequency spectrum, the bandwidth is increased. Althouqh 
the "creeD" is relatively small for small values of A6, it must be 
considered. 

This creeping bandwidth may be controlled in several ways. If 
the bandwidth requirements of the filter are not too stringent or 
the "0" of the "circuit" is not required to be too larqe, this 



24 



widening may be neglected or taken into account when designinq. This 
was the case in this project. Even though the upDer partial s of the 
piano are not integer multiples of the fundamental, they are some- 
where in the vicinity of the pure harmonic frequency and a relatively 
"relaxed-0" filter could be implemented. 

Usinq this reasoning as a basis, A9 was selected such that as the 
samDling frequency was increased, the filter would continue to nass 
only the one desired partial of the complex sound generated. This 
is well illustrated in the computer output from program two. 

However, if the filtering requirements and "0" are such that no 
amount of variance in the bandwidth can be tolerated, the problem 
becomes more complex and much more interesting. Fiq. 8a illustrates 
two tuneable digital filters in cascade and separated by an inverse 
digital filter. The inverse filter merely changes the output of 
filter #1 into an analog form in order that the resultant wave may be 
sampled by filter #2. The shaded area of Fig. 8b depicts the spectral 
output of the second filter. It would be desireable to keep this 
interval constant as it was swept up and down the frequency snectrum. 

Let 6 be a constant and represent this interval. Then 

f 2 - f 3 = 6 (33) 

f 3 = f 2 - 6 (34) 

but from previous discussion 

f 3 = e 3 f s2 ' 2lT (35) 

f 2 = e 2 f sl / 2-tt (36) 

substituting (37) and (36) into (35) 



25 



3 f s2 / 2-rr = 9 2 f $1 / 2tt - 6 (37) 

f s2 = 2tt / e 3 (e 2 f s l /2 ^- 6) (38) 



which can be reduced to 



where 



f s2 = mf sl - b (39) 



m = e 2 /e 3 (40) 



b = 2-n-5/e 3 C4.1) 

These equations illustrate quite clearly that a linear relationship 

between f , and f « can be established that results in a definite 
s I S^l 

and invariant bandwidth. To calculate the center frequency f of 

this passband it is clear from Fig. 8b that 

f c = f 2 - 6/2 (42) 

= e 2 f sl /27t - 6/2 (43) 

which results in 



f c - af sl - d (44) 



where 



a = e 2 /2 (45) 

d = 6 /2 (46) 

Equation (44) like (39) clearly demonstrates the fact that the center 
frequency of the passband is a linear function of the samplinq fre- 
quency f i . Therefore it is quite possible to implement a tuneable 
digital filter that has an invariant bandwidth by simply adding 
another filter with a different samplinq frequency which is linearly 

related to f , . 
si 



26 



The computer programs in the back of this paner were written to 
test the above theory of a possible tuneable fixed-coefficient diqital 
filter. The first one takes as inputs the locations of the poles and 
zeroes in the complex z-domain and calculates the proper filter coeffi- 
cients. This can be done for filters up to and including n = 50. 
Filter responses for five different sampling intervals were then plotted 
via the DRAW sub-routine. The second program calculates the samoling 
frequency required for filterinq the desired partial s of the fundamen- 
tal frequencies of the piano. The key number, the fundamental fre- 
quency, tb'e partial to be filtered, the maximum allowable bandwidth 
to filter the partial, theta, delta theta, the required sampling 
frequency and the resultant bandwidth of the filter are all calcu- 
lated, outputed and tabulated via this proqram. 

Not only does this output clearly illustrate the nrocedure in- 
volved in tuning the piano, but it also gives a good insight into the 
tuneable filter. That is, it demonstrates that once the coefficients 
of the filter have been set (i.e., theta and delta theta), the filter 
can be made to sweep the freguency spectrum by chanqing the samplinq 
interval . 

The output from orogram number one, however, offers the most con- 
clusive nroof that a tuneable digital filter is realizable. As ex- 
plained previously the program calculates the coefficients for an n 
order filter. It then takes these coefficients and using them in the 
eguations from sampled-data theory, computes the frequency resoonse of 
the filter for five different sampling frequencies. The granhs on 
pages 46 - 51 are self explanatory and offer convincing evidence that 
the previously derived equations and theory for a tuneable digital 
filter were correct. 

27 



D. ERRORS THAT MUST BE CONSIDERED 

Since in the process of tuning the designed filter, the sampling 
interval was allowed to move over quite a large range, care had to 
be taken in avoiding the well known "aliasing" or "foldover" distor- 
tion problem. This distortion results when the samoling frequency 
is not at least twice as high as the highest frequency comnonent found 
in the total input signal. This problem can be overcome by employing 
wide-band samDled data filters and "pre-warping" the frequency being 

Q 

filtered [Ref. EG-15 and EW-48] . This approach was not used because 
of a much easier solution that resulted from the restrictive parameters 
of the problem. 

The highest frequency from the piano that was required to be fil- 
tered and measured was the fundamental of C8 (4186.009 Hz). Two pre- 
cautions were taken to insure that foldover distortion did not occur. 
The circuitry involved with the microphone input was designed to act 
as a low-pass filter with the upper cut-off frequency just above the 
C8 fundamental frequency. Secondly, in the design of the filter, 
f was set equal to six times the value of the hiqhest input frequency 
possible from the "low-pass filter." e was then calculated from 
(31) and "hard-wired" into the z-domain. This resulted in a safety 
factor of at least five over the entire frequency spectrum to be 
measured. 



9 

By wide-band sampled-data filter is meant a filter whose fre- 
quency range approaches half of the samoling frequency. The cited 
reference gives an excellent account of this type of filter and 
its possible implementation. 



28 



The angle e c was computed as being equal to 30°. A6 was made 
equal to 9° in order to make the bandwidth of the filter as wide as 
possible and yet still keep it within the maximum allowable range. 
Ten coefficients were called for in order that the filter would have 
as sharD a response as possible but yet still steer clear of a delay 
and coefficient accuracy problem caused by the use of too many coeffi- 
cients. The reader is referred to the computer outputs previously 
mentioned for the tabulation of all coefficients. With the mention 
of coefficient accuracy it seems aopronriate to say somethinq about 
round-off errors, word length requirements, coefficient accuracy and 
associated problems. 

When a digital filter is realized with a digital arithmetic 
element, as is the case of the proposed tuner under study, additional 
considerations are necessary to describe the performance of the filter. 
There are three obvious degradations: 

1. quantization of the input, 

2. quantization of the coefficients of the difference 
equation, 

3. quantization of the results of the comoutations. 

All three types have been thoroughly investigated in Refs. EK-21 , EM-27, 
ET-44, EK-19 and ET-45, and only the various results that affect this 
problem will be presented here. 

The two prime considerations that enter into the selection of the 
input quantization size, q, are a minimum detectable level of the sig- 
nal, x TH , and a saturation level, x^ AT . From Ref. EW-43 it was found 
that 

q = x TH /x SAT . (47) 



29 



The A/D converter is then required to have a minimum accuracy of N 

bits, which is determined by: 

2" N = q (48) 

N = log 2 (x SAT /x TH ). (49) 

One word time is then defined by N data-bit times plus one sign-bit 
time. The speed of the converter, although no problem in this pro- 
posed design, may be calculated from the following formula where 
T = sampling interval 

S = (2/T) (1 + log 2 (x SAT /x TH )). (50) 

The third quantization problem mentioned is usually set equal to 
the quantization of the inout and was done so in this design. See 
Ref. EW-48 for further amplification. 

The second type of quantization error listed appears to be the 
most important one to this writer. It is quite easy to understand 
and a thorough investigation is conducted by Knowles and Olyacato in 
Ref. EK-19. This quantization of the coefficients chanqes them slight- 
ly, resulting in a new, slightly different filter response. This 
happens only once when the filter is first designed. One important 
aspect of this problem is that the more coefficients that are used, 
the more accurate they must be and therefore the more this error enters 
into the problem. This is why a filter of the order of only n = 10 
was chosen. Kaiser, in Ref. EK-21 , also brings out the fact that as 
the sampling frequency is raised the coefficients must become closer 
and closer to the "ideal" ones; that is, ones that would require an 
"infinite" number of bits for their implementation and storage. This 



30 



was the reason that the sampling frequency was set to be onlv six 
times the value of the highest input frequency instead of ten to twelve 
times as is done in most samnled-data networks. 

It should be Dointed out here that, although the sampling 
frequency was kept as low as possible and the order of the filter 
was kept relatively small, a great deal of difficulty was encountered 
with the coefficient accuracy problem. 

The coefficients given in the comnuter output section were calcu- 
lated using complex double-precision arithmetic on the IBM/360/67. 
These same numbers were obtained by two completely different algorithms 
so there is little doubt that they are correct. However, when the coef- 
ficients were used to obtain the frequency response of the desired 
filter, the undesirable effects shown in Granhs 1 and 2 in Anpendix A 
were obtained. This was, however, not necessarily due to inaccurate 
coefficients. Two possible inaccuracies could have been generated by 
the nature of the library sub-programs used. First, in order to take 
the real part of the complex coefficients, the precision had to be re- 
duced from double to single. This naturally involved some tyne of 
round-off error. This necessitated change in precision resulted in 
the use of the single-precision sine and cosine library tables and 
could have introduced another round-off error. Short of actual imole- 
mentation of these coefficients in a digital filter, any further 
attempt to obtain the frequence response by this method was imDossible. 

In order to bypass this problem Equation (13) was programmed. 
As is readily apparent by referring back to (13), this equation 
greatly reduces the probability of round-off errors. This reduction 
is caused by the knowledge of the precise locations of the ooles 



31 



and zeroes and the requirement of having to introduce only one phasor 
calculation. This is unlike the previous example where the order of 
the filter determined how many different phasors had to be solved. 
The results illustrated in Graphs 3-6 in Appendix A conform to all 
theory that has been presented. With this discussion, this author 
feels that if the tabulated coefficients were actually implemented 
in a digital filter the frequency response of Graohs 3-6 would be 
realized. 



32 



III. A PROPOSED DESIGN FOR A DIGITAL PIANO TUNER 

A. INTRODUCTION AND OVER-ALL VIEW OF THE MACHINE 

The third and final section of this paper is devoted to the 
actual implementation of the proposed tuner. Fig. 9 illustrates 
the general flow of signals and information in block diagram form. 
In actual operation the key to be tuned is selected on the front 
of the control panel by the rotation of a 12-position switch repre- 
senting the twelve notes of one of the equal -tempered octaves. 
Another rotary switch is used to select the appropriate frequency out- 
put of the Reference Frequency Generator Module (RFGM) as will be 
explained later. The RFGM is nothing more than a very stable, crys- 
tal-controlled I.C. pulse-train generator. The output frequency of 
this module had to be at least four times as high as the desired fre- 
quency to be measured. This was required in order that the intervals 
generated by the ring counter would be quarter periods and the subse- 
quent error count could therefore be read in cycles. 

The Time-Base Generator Module (TBGM) was added to provide the 
error count readout directly in hertz. This module acts as a real— time 
clock (RTC) and was provided with the capability of producing timing 
pulses every 5 or 10 seconds. 



Actually the use of the word "key" here is a misnomer. All keys 
have more than one string in order to increase their output volume. The 
normal modern-day piano has, starting at the upper octaves and working 
towards the extreme bass, 60 keys with three strings, 13 keys with two 
strings and 10 keys with one string. Some of the larger concert pianos 
will have more but all pianos have the same 88 keys. The different strings 
are each tuned separately. This is achieved by insertinq a tuning wedge 
between the strings in such a way that only one is allowed to vibrate. 
Extreme care must be taken that the multiple-stringed keys are tuned 
correctly if undesirable "beats" are to be avoided. 

33 



The Sign-Detection and Pulse-Shaping Module tests the left-most 
bit of the output register of the digital filter and when a positive- 
going change occurs, that is, when the bit changes from a "0" to a 
"1" a "one-shot" is triggered. Since an output pulse from the one-shot 
occurs only when the filtered partial to be measured crosses the 
"zero axis," the pulse is in synchronization with the unknown frequency 
of the filtered partial. This pulse from the one-shot is fed into 
the Frequency Deviation Detector and Counter Module (FDDCM). 

The Frequency Deviation Detector and Counter Module is a logic 
circuit that is capable of measuring the beat frequency between the 
reference frequency and the unknown input. This beat frequency or 
error count is displayed on the control panel by a series of digital 
display lights. The FDDCM also determines whether or not the beat 
frequency is sharp (fast) or flat (slow) with respect to the reference. 
This enables the tuner to know in which direction to turn the tuning 
pin in the piano. For the laymen, this sharp-flat meter would be 
all that would be required for successful tuning. However, most pro- 
fessional tuners like to sharpen or "stretch" the upper octaves by 
about five cents per octave to add a certain amount of "brightness" 
to the sound generated [Ref. MH-8]. This requires some form of exact 
frequency measurement, which the FDDCM does to a very precise degree. 

As mentioned previously, the input microphone and associated cir- 
cuitry was designed to act as a low-pass analog filter with an UDper 
cut-off point at approximately 5 KHz. 

The Digital Filter Module multiplies the sampled input by the appro- 
priate coefficients. This involves the implementation of shift regis- 
ters and some associated logic. 



34 



The Display and Control Module (DCM) houses the various selection 
switches and lights for display. Four 10-counters were properly 
implemented and decoded to display the error in the incoming freguency 
from the piano. 

B. THE POWER SUPPLY 

The power supply had to be designed to provide more than an 
amoere of current at 3.6 volts DC to the various logic chins. This 
had to be supplied with a very low ripple factor in order to Drevent 
confusion between the ripple and the actual timing pulses emoloyed in 
the circuit. It had to also provide other lower-current supplies for 
additional circuitry at +6, -6, and +12 volts DC. In addition to 
the various bias requirements it had to meet, it was also desirable to 
produce a 60-Hz signal for the Time-Base Generator Module and an AC 
voltage to the four digital— display lights. 

In order to meet all of these requirements from a single trans- 
former, eleven diodes had to be used. See Fig. 10 for the schematic 
of this module. 

The +12-volt supDly was obtained from a voltaqe doubler consisting 
of Dl , d2, CI and C2. The full-wave rectifier made up of D3, D4 and 
C5 provided the -6 volts, while a second full-wave rectifier, consisting 
of D5 and D6 fulfilled the +6 volts bias requirement. The +6 was 
reduced by D9, D10 and Dll to provide the +3.6 volts required by the 
I. C. chips. 

C. THE ANALOG TO DIGITAL CONVERTER 

The A/D module is illustrated in block diagram form in Fig. 11. 
The operation of the ramp converter is initiated by means of a samnle 



35 



gate pulse which is applied to the output of the ramo generator. The 
ramp generator produces the triangular waveshape shown in Fig. 12, wave- 
form A. As this rising voltage becomes equal to or qreater than the 
zero reference applied to comparator No. 1, a nulse is produced which 
is used to set a control flip-flop. When the rising voltaqe becomes 
,equal to or greater than the analog voltage applied to comparator No. 2, 
a pulse is produced to reset the control flip-flop. The result of this 
operation is the production of a control pulse, the duration of which 
is proportional to the amplitude of the input analog signal. This con- 
trol pulse is then used to gate clock pulses (which are a periodic 
train of timing pulses) into a counter which starts at a count of zero. 
The number of pulses allowed to enter the counter is proportional to 
the pulse width of the control pulse. Therefore, the final content of 
the counter is proportional to the amplitude of the analog signal. 
After the ramp voltage reaches a maximum, it is returned to its initial 
value. The trailing edge of the ramp is used to generate a reset pulse 
which reads out the contents of the counter into an output register 
and resets the counter to zero. All of the associated waveforms are 
illustrated in Fig. 12. The "transfer" occurs when the seven-bit 
counter is read-out into the read-out gates. 

The seven-bit counter illustrated in Fig. 11 had in reality eight- 
bits. The left-most bit, the sign-bit, was drooped from the above 
discussion for the sake of clarity. The seven-bit counter (including 
the eighth-sign-bit) was implemented because it produced a ^jery high 
and desirable degree of accuracy. That is, it was capable of dividing 
the input signal into 628 levels (2 7 = 628). 



36 



D. THE FREQUENCY DEVIATION DETECTOR AND COUNTER MODULE 

If two frequencies are superimposed on an oscilloscone screen, 
the relation between them can be determined by count inq the number of 
times they move in and out of synchronism. In reality this nothinq 
more than determines a beat frequency, which will qive the deviation of 
the second from the first, if the first is considered to be the refer- 
ence. In existinq analoq systems the attemnt to realize this beat 
frequency at low frequencies and with small errors can be quite complex 
and cumbersome. For example, an error of 0.01% at 400 Hz would produce 
a beat of one twenty-fifth of a cycle per second. Althouqh accuracy 
this hi qh is not demanded in the tuninq of the piano, a reasonable deqree 
of accuracy is required and can be realized with a saving in size, 
cost and required skills of an operator by the use of diqital circuits. 
Refer to Fig. 13 for the loqic diaqram of the desiqned FDDCM. 

The crystal -control led reference frequency from the RFGM is fed 
into the phase qenerator, a simple four-staqe ring counter (See Fig . 14), 
to produce the required timinq intervals. Four intervals, A thru D, 
were provided to allow for the Dolarity of the errors to be calculated. 
The one-shot (mono-stable multivibrator) produces a nulse that is in 
synchronization, as explained previously, with the unknown frequency 
to be measured. This pulse, desiqnated f , was ANDed separately with 
each of the four phases from the ring counter. This gives the outnut 
of the coincidence detector as (AF, BF, CF and DF). These signals 
provided the basis for the remaining logic. The error-count detector 
was an RS flip-flop which produced an error count whenever the unknown 
had cycled from coincidence with phase A to nhase C. The recordina of 



37 



an error count occurred only when the output of this RS flip-flop 
changed from "0" to the "1" state. Note that repeated Dulses on the 
set input can produce no further error counts. 

The polarity, that is whether or not the unknown is fast or slow 
compared to the reference, was determined by the remaining logic. 
Figs. 15, 16 and 17 illustrate the various waveshapes for a "sharp," 
"flat" and perfectly tuned unknown input. If the input is perfectly 
"tuned" to the reference it will aDpear to remain stationary when 
plotted against the reference with respect to time. However, if 
it were sharp or flat, it would tend to drift to the right or left, 
respectively, and thus cause an error count to be generated. 

Once again, only off-the-shelf Fairchild Semi -Conductor Micro- 
Logic Chips were employed in all design work. 

E. THE REFERENCE FREQUENCY GENERATOR MODULE 

The RFGM works on the same basic principle as the pitch reference 
found in Ref. EL-22. That is, it takes a set, crystal -control led 
freguency and uses it to drive a chain of JK flip-flops. This chain 
divides the crystal frequency output into the desired reference fre- 
quency outputs. 

The power supply used in the cited reference was deleted in favor 
of the one already discussed. This was only for the sake of simplicity 
and in an attempt to avoid needless repetition of components. The 
speaker and associated circuitry were also dropped for the same reasons, 

The basic block diagram of the circuit appears in Fig. 13. When 
one of the crystal oscillators is switched on, the output wave is taken 
by the input of the Logical Schmitt Trigger, illustrated in Fig. 23, 



38 



and made into a square wave of the same frequency. Take for example, 

the switch S, as being in the position indicated in Fig. 19. Then 

the output of the Schmitt Trigger is a 34.29 MHz square wave and is fed 

into the chain of JK flip-flops. With the switch in this position there 

are 12 flip-flops in this chain when tuning octaves five, six and seven. 

With no feedback, the chain will divide by 4096 (2 = 4096). The 

resulting output frequency of the chain would therefore be 8371.9 Hz. 

Notice that this frequency is four times the frequency of the desired 

fundamental of C7, the desired first partial of C6 and the third partial 

of C5. It is therefore the proper reference frequency required to 

input to the FDDCM when tuning these notes. 

Now if the frequency output from the Drevious example were buffered 

and fed back into the outputs of flip-flops one, two, five and six, 

the chain would now divide by 3866. This is because extra counts 

exactly equal to the difference between the desired divisor of 3866 

and 4096 have been added. In arithmetic form the preceedinq is 

illustrated by 

4096 - 3866 = 130 

1 ? 4 "5 6 

2+2+2+2+2 = 130 

When this division by 3866 is carried out on the input frequency of 34.29 

MHz, the resulting output was calculated to be 8869.6 Hz. This is four 

times the desired fundamental of C7#, the desired first partial of C6# 

and the desired third partial of C5#. Once again the required 



39 



reference frequency to tune these notes has been obtained. This same 
procedure was carried out for the 36 note 
and the results are displayed in Table 2. 



procedure was carried out for the 36 notes in the upper three octaves 

11 



NOTE 



C7-C6-C5 

C7#-C6#-C5# 

D7-D6-D5 

D7#-D6#-D5# 

E7-E6-E5 

F7-F6-F5 

F7#-F6#-F5# 

G7-G6-G5 

G7#-G6#-G5# 

A7-A6-A5 

A7#-A6#-A5# 

B7-B6-B5 



DIVISION 


DESIRED PARTIAL 


REFERENCE 


RATIO 


FREQUENCY 


FREQUENCY 


4096 


2093.0 


8371.9 


3866 


2217.5 


8870.0 


3650 


2349.3 


9397.2 


3444 


2489.0 


9956.0 


3250 


2637.0 


10548.0 


3068 


2793.8 


11175.2 


2896 


2959.9 


11839.6 


2734 


3135.9 


12543.6 


2580 


3322.4 


13289.6 


2436 


3520.0 


14040.0 


2298 


3729.3 


14917.2 


2170 


3951.1 


15804.4 



TABLE 2 

Reference frequency output data for 
the upper three octaves 



As is illustrated in this table all of the upper three octaves of the 
piano can be tuned with the 34.29 MHz frequency setting and the chain 
of twelve dividing flip-flops. 

As explained previously it was desired to correctly measure and 
tune the fourth partial s of the notes in the fourth octave. This octave 
of frequencies can be obtained by dividing the third Dartials of the 



Although the reference frequency output is not exactly four 
times the value of the partial being measured, it is accurate to 
± 0.5 cent, making it twice as good as the best tuning fork 
available. 



40 



fifth octave by two. Since it has already been shown how the RFGM is 
able to generate the correct frequency for the fifth octave, the only 
requirement is to divide this set of frequencies by two in order to 
obtain the correct set of frequencies for the fourth octave. That is, 
when tuning the fourth octave, another flip-flop must be switched in 
from the control panel. This is done by simply rotating the seven- 
position switch marked "octave" to the 4th position. 

It was also desired to measure and tune the fourth partial s of 
the lower three and one-third octaves. It would have been highly 
desirable to simply continue adding divide-by-two flip-flops in the 
dividinq chain to decrease the reference frequency to its desired 
value. Had this been done, however, it would have been required to 
filter and measure the ninth partials of the third octave, the eighth 
partial s of the second octave and the seventh partials of the first 
octave. According to Plomb's criterion this would have presented no 
problem. However, these specific partials are unfortunately sup- 
pressed by manufacturers because of the shrillness they tend to add 

1 2 
to the music when present [Ref. EW-22] . Clearly, another approach 

was indicated. 

This observation required that a second crystal with a resonant 

frequency of 6.7171 MHz be added. When this frequency was divided by 

the basic chain of twelve flip-flops, a set of frequencies was 

produced that was four times the desired partial frequencies of the 

fourth octave. Therefore, the required reference frequency had once 



1 2 
This is done by having the hammer hit the strinq between one- 
seventh and one-ninth of the distance from the one end of the string 
to the other. 



41 



again been obtained. The first and second octaves can also be tuned 
from this crystal by switching one more flip-flop for the second oc- 
tave and two more for the first. Once again this is achieved by the 
rotary switch on the face of the Display and Control Module. 

F. THE TIME BASE GENERATOR MODULE 

The TBGM, illustrated in Fig. 19, consists of seven JK flip-floDS 
and one dual two-input gate package (Fairchild Semiconductor RTyL 9914). 
Incoming 60-Hz clock pulses from the power supply are fed to FF1 and 
FF2 which divides the input by three. The resulting 20-Hz signal 
provides clock pulses at 0.05-second intervals (T = 1/f). FF3 divides 
the 20-Hz signal by two to provide clock pulses at 0.10-sec intervals, 
and FF4, FF5 and FF6 are connected in a divide-by-five circuit config- 
uration to obtain the 0.50-second timing interval. The 2-Hz output 
of FF6 is divided by FF7 for the 1 .0-second clock Dulse. The first 
half of the dual two-input gate is used as an inverting amplifier to 
boost the divide-by-five input signal level and the other half is 
used to amplify the final clock-oulse output. This outnut chain of 
pulses is fed into a divide-by-ten counter illustrated in Fig. 20. 

The divide-by-ten module was designed using the set and clear of 
the basic JK flip-flop as gates. It was able to feedback directly to 
the binary divider without any extra parts and inhibit counts 11 
through 16. Although the circuit is quite difficult to decode, it 
provides a s/ery simple, stable and cheap divide-by-ten package. 
The output from this circuit provides the actual five or ten-second 
timing pulse. When the pulse occurs it gates the FDDCM and halts all 
counting. 



42 



In order to ensure that the 60-Hz signal obtained from the power 
supply had a relatively sharp rise and fall time, a Logical Schmitt 
Trigger was implemented between the power supply and the input to the 
TBGM. It is illustrated in Fig. 21 and as shown emnloys a Fairchild 
RTyL 9914 gate. The series resistance limits the Deak voltage at the 
gate input to two to three and one-half volts. The capacitor acts 
as a filter and the diode clamps the negative portion of the wave to 
ground. 

G. THE TUNEABLE DIGITAL FILTER MODULE 

The implementation of this module might have presented a real 
problem if the use of a tuneable filter had not proven feasible. It 
would have then been necessary to generate a new set of coefficients 
for each octave to be tuned. Fortunately, as has already been dis- 
cussed in some detail, this was avoided. 

The filter used has the same characteristics as the one discussed 
in Ref. ET-43, except that the sampling time could be varied by the 
tuner from the CDM. This Fixed-Coefficient Diode-Array Digital Filter 
was employed because of its extremely small size and simplicity of 
implementation. It uses the table-lookup method of product generation 
where the diode-array is used to store the multiplication table. A 
diode-array mechanization is illustrated in Fig. 23. 

The input data from the A/D converter is shifted in serially to 
load the shift reqister. The contents of this register addresses the 
product, which is stored, in the diode array. The product is loaded 
into the output reqister which supplies the input to the next chip or 
array. Each chip delays the information the required interval T. 



43 



The schematic of Fig. 22 can be implemented on a single MOS chin. 
For this proposed tuner it would take 20 of these chins; one for each 
numerator and denominator coefficient. Reference ES-38 discusses 
recent developments in this field and also the possibility of putting 
many of these arrays on a single chip. 

H. THE DISPLAY AND CONTROL MODULE 

The DCM consists of a set of digital display lights, a meter indi- 
cating flat, zero and sharp, and two sets of selection switches. The 
selection switches control the number of flip-flops in the RFGM dividing 
chain and the sampling interval of the A/D. The meter is a three - 
terminal galvonometer and will tell the tuner whether the partial 
being measured is sharp or flat with respect to the reference fre- 
guency. The display lights will measure a frequency deviation up to 
999.9 Hz. There are four lights on the panel and each one is used 
to decode a "10" counter. The counter and the required decoding 
scheme are illustrated in Fig. 24. The output from the first counter 
is used as the trigger for the second. The output from the second is 
used as the input for the third and the output from the third is used 
as the input for the fourth and final stage. Thus, a "1000" counter 
has been effectively achieved. When employing the usual 10 second- 
timing interval, the counter becomes a "100" counter that counts by 
tenths. 



44 



IV. CONCLUSION 

From the study of the physics of the piano, it has been shown 
why it was necessary to measure and tune the eigen-partials of the 
complex sound produced by the steel -stringed piano. It was also 
demonstrated how this filtering and frequency measurinq could be 
achieved with a tuneable fixed-coefficient digital filter and several 
additional I.C. modules. The remaining portion of the work was then 
devoted to the possible implementation and construction of these 
modules. Based on the sound and music theory presented this pro- 
posed design seemed to offer the best electronic tuner to date. 
Unfortunately time did not allow for the actual interfacing of the 
modules and testing of the tuner as a single unit. 

However, it should be stated that this author in no way advocates 
this machine as a panacea for the complex problem of tuninq equal- 
tempered key-board instruments. Music and its desirable qualities 
are things which surpass scientific measurement because, contrary to 
all attempts to define them otherwise, they remain purely subjective 
quantities. A piano tuner who emoloys his well-trained ear is able 
to take the tastes of the period into account and tune accordingly. 
This machine or any other can only give him a better or more per- 
ceptive "ear." 



45 



APPENDIX A GRAPHS 



20 



D 
E 
C 
I 
B 
E 
L 
S 



10 



-10 



-20 



-30 



-40 




200 



400 
HERZ 



600 



800 



Graph 1. Illustrating the Results of Coefficient 
Inaccuracy on the Frequency Response of a 
Digital Filter. 



46 



30 



20 



10 



-10 



-20 



-30 




aoo 



Graph 2. Illustration the Results of Coefficient 
Inaccuracy on the Frequency Response of a 
Digital Filter. 



47 




200 401 

HERZ 



600 



f.=2.750 KHz n=10 



Graph 3 . Frequency Response of the Designed 
Digital Filter. 



48 



50 



40 



30 



20 



'1 



200 400 
HERZ 



600 



f s =4.125 KHz 



n = 10 



Graph 4. Frequency Response of the Designed 
Digital Filter. 



49 



50 





E 

C 40 

i 

a 

E 
L 
S 

30 



20 



i 
i 

f ! 



400 



600 800 
HERZ 



1000 



f -P.1C0 kHz 



n=10 



Graph 5 frequency Response of t'r e Designed 
Digital Filter. 



50 




600 800 
HCRZ 



1000 



f s =9.281 KHz 



n=10 



Graph 6. Frequency Response of the Designed 
Digital Filter. 



51 



APPENDIX B FIGURES 



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55 



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57 



FIRST 

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THE FOLLOWING TABLE WAS GENERATED USING THETA- 30.0 ANO OELTA THETA- 9.0 DEGREES. 

•rev rnnni.fiTii P1RTIAI TO PARTIAL LOWER UPPER LARGEST SAMPLING RESULTANT 
En* fEfSueEcy BE TUNED FREQUENCY FREQUENCY FREQUENCY BANDwToTH FREQUENCY BANDWIDTH 



1 


27 


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29 


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4.0 137.5000 110.0000 165.000 55.0000 165C.000 41.2500 
4*0 145 # 6750 116.5400 174.810 58.2700 1748.100 43.7025 
i*C 1*4:3400 123.4720 185.208 61.7360 1P52.080 46.3020 



1 OCTAVE 

4.0 163.5149 130.8120 196.218 65.4060 1962.179 49.0545 

4*0 173.2400 138.5920 207.889 69.2960 2078. 8B0 51.9720 

<,.C 183.5400 146.8320 220.248 73.4160 2202.479 55.0620 

4*C 191.9550 153.5640 230.346 76.7820 2303. 45e 57.5864 

4^0 206.0149 164.8120 247.218 82.4060 2472.179 61.8045 

4.0 218.2700 174.6160 261.924 87.3079 2619.237 65.4809 

4.0 231.2450 184.9960 277.494 92.4979 2774.937 69.3734 

4.0 244.9950 195.9960 293.994 97.9979 2939.937 73.4984 

4.C 259.5649 207.6519 311.478 103.8258 3114.779 77.8695 

4.C 27CO0OO 220.0000 330.000 110.0000 3300.000 82.5000 

4.0 291.3499 233.0799 349.620 116.5398 3496.198 87.4049 

4.0 308.6748 246.9398 370.410 123.4699 3704.096 92.6024 

2 CCTAVE 

4.0 327.0298 261.6238 392.436 130.8118 3924.356 98.1089 

4.0 346.4797 277.1836 415.776 138.5920 4157.754 103.9438 

4.C 367.C798 293.6638 440.496 146.9318 4404.953 110.1238 

4.0 'PP. 9099 311.1277 466.692 155.5642 4666.918 116.6729 

4.0 412.0349 329.6277 494.442 164.3142 4944.418 123.6104 

4.C 436.5349 349.2278 523.842 174.6140 5238. 41 e 130.9604 

4.C 462.4949 369.9958 554.994 184.9978 5549. 037 138.7484 

4.0 489. c 949 391.9958 587.994 195.9978 5879.937 146.9984 

i.C 519.1299 415.3037 622.956 207.6521 6229.555 155.7399 

4.0 550.0000 440.0000 660.000 220.0000 660C.OOC 165.0000 

4.C 582.7048 466.1638 699.246 233.0813 6992.453 174.8113 

4.0 617.3547 493.8835 740.826 246.9421 7408.254 195.2062 

3 CCTAVE 

4.C 654.C649 523.2520 784.878 261.6260 7848.777 196.2193 

4.0 6=2.9548 554.3638 831.546 277.1819 8315.453 207.8863 

4.C 734.1599 5e7.3279 880.99? 293.6638 8809. 91P 220.2479 

4.0 777.8149 622.2520 933.378 311.1260 9333.777 233.3443 

4.0 824.0698 659.2556 998.884 329.6292 9888.836 247.2208 

4.0 873.0699 698.4558 1047.684 349.2278 10476.836 261.9207 

4.0 924. "849 739.9878 1109.982 369.9939 11099.316 277.4951 

4.0 979.0897 783.9=17 1175.988 391.9953 11759.675 293.9968 

4.C 1038.2599 830.6077 1245.912 415.3040 12459.113 311.4775 

4.0 110c. 0000 eso.cooo 1320.000 440.0000 13200.000 330.0000 

4.C 1165.4099 932.3279 1398.492 466.1638 13984.918 349.6228 

4.C 1234.7097 987.7676 1481.652 493.9840 14816.512 370.4126 

4 CCTAVE 

4.C 1308.1299 1046.5039 1569.756 523.2520 15697.555 392.4385 

4.0 1385. =143 1108.7314 1663.097 554.3657 16630.969 415.7739 

-..0 1463.3240 1174.6592 1761.989 587.3296 17619.887 440.4968 

4.0 1555.6349 1244.5078 1866.762 622.2539 18667.613 466.6902 

4.C 164P.1396 1319.5117 1=77.769 659.2559 19777.672 494.4417 

4.C 1746.1389 1396.9111 2095.367 698.4556 20953.664 523.8413 

4.0 1349. =695 1479.9756 2219.963 739.9976 22199.633 554.9907 

4.C 1959.0744 1567.9795 2351.969 783. 98= 7 23519.691 587.9922 

4.0 2076.5247 1661.2197 24=1.830 830.5099 24918.293 622.9573 

4.C 220CC000 1760.0000 2640.000 680.0000 26400.000 660.0000 

4.0 2330.8191 1864.6553 2796.983 932.3276 27969.824 699.2454 

4.0 2*69.4141 1975.5312 2963.297 987.7656 29632.969 740.8240 

5 CCTAVE 

3.C 2093. 0039 1569.7529 2616.255 1046.5020 25116.043 627.9009 

3.0 2217.4600 1663. 0950 2771.825 1109.7300 26609.516 665.2379 

3.C 2349.3193 1761.9895 2936.649 1174.6597 23191.828 704.7957 

3.0 2489. C156 1866.7617 3111.270 1244.5078 29868.187 746.7046 

3.0 2637. C195 1977.7646 3296.274 1318.5098 31644. 23C 791.1057 

3.C 2793. e232 2095.3674 3492.279 1396.9116 33525.875 838.1467 

3.0 2=59. =551 2219.9663 3699.944 1479.9775 3551°. 430 887.9854 

3.0 3135. =639 2351.9729 3919.955 1567.9319 37631.531 940.7931 

3.0 3322.43*5 24=1.8267 4153.043 1661.2163 39869.199 996.7297 

3.0 3520. COOO 264C.0000 4400.000 1760. OOCO 42240. OOC 1056.0000 

3.0 3729.3115 2796.9836 4661.637 1864.6531 44751.730 1118.7932 

3.C 3951. C674 2963.3005 4938.832 1975.5315 47412.797 1185.3196 

6 CCTAVE 

1.0 2093. 0039 1046.5020 3139.506 2093.003= 25116.043 627.9009 

l.C 2217.4619 1108.7310 3326.193 2217.4619 26609.539 665.2383 

1.0 2349.3179 1174.6589 3523.977 2349.317= 28191.812 704.7952 

1.0 2489. C156 1244.5078 3733.523 2489.0156 29868.187 746.7046 

l.C 2637. C195 1318.5098 3=55.529 2637.0195 31644.230 791.1057 

1.0 2 7 93.6257 1396.9128 4190.738 2793.3254 33525.906 838.1475 

1.0 2=59. =556 1479.9778 4439.930 295=. 9519 35519.465 887.9863 

l.C 3135.9639 1567.9819 4703.945 3135.9634 37631.531 940.7881 

l.C 3322.4380 1661.2190 4983.656 3322.4373 3=869.230 996.7307 

1.0 3520.0000 1760. COOO 5280.000 3520.0000 42240. OOC 1C56.C000 

l.C 3729.3096 1864.6548 5593.961 3729.3062 44751.699 1118.7922 

l.C 3=51.0659 1975.5330 5926.598 3951.0647 47412.766 1185.31=1 

7 CCTAVE 

C.C 2093.0049 CO 4186.009 4186.0079 25116.055 627.5011 

3.0 2217.4609 0.0 4434.922 4434.921= 26609.531 665.2380 

CO 2349.3179 0.0 4696.633 4698.6329 28191.812 704.7952 

0.0 2489. 0159 CO 4978.031 4978.0312 2986e.l87 746.7046 

0.0 2637. C210 0.0 5274.039 5274.0391 31644. 25C 791.1062 

C.C 2793.6259 0.0 5587.649 5587.6484 33525.910 638.1477 

CO 2=59.9548 O.C 5919.906 5919.9062 35519.430 687.9854 

CO 3135. =639 CO 6271.926 6271.9258 37631.531 94C7881 

O.C 3322.4380 CO 6644.875 6644.9750 39869.230 996.7307 

0.0 352CCCOC 0.0 7040.000 704C0000 42240.000 1056. COOO 

CC 3729.3099 0.0 7458.617 7458.6172 44751.699 1118.7922 

0.0 3951.065= CO 7902.129 7902.1289 47412.766 1185.3191 

0.0 4186. C078 CO 8372.016 8372.0156 50232.066 1255.8015 



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BIBLIOGRAPHY 



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91 



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92 



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93 






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Electronics World , v. 81, n. 1, p. 49-55, January 1969. 

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Annual National Electronics Conference, 1968. 

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94 



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1. Defense Documentation Center 9n 
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Naval Postgraduate School 
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95 



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DOCUMENT CONTROL DATA -R&D 



(Security classification of title, body of abstract and indexing annotation must be entered when the overall report Is classified) 



(.ORIGINATING ACTIVITY (Corporate author) 

Naval Postgraduate School 
Monterey, California 93940 



2«. REPORT 5ECURI TY CLASSIFICATION 



Unclassified 



2fc. GROUP 



3 REPORT Tl TLE 



An Integrated-Circuit Piano Tuner for the Equal -Tempered Keyboard Employing 
a Tuneable Fixed-Coefficient Digital Filter 



4, DESCRIPTIVE NOTES (Type of report and. inclusive dates) 

Master's Thesis; June 1969 



5. au THORIS) (First name, middle initial, last name) 



Michael William Haqee 



6. REPORT DA TE 

June 1969 



la. TOTAL NO. OF PAGES 



96 



76. NO. OF P.EFS 



75 



»a. CONTRACT OR GRANT NO. 



6. PROJEC T NO. 



9a. ORIGINATOR'S REPORT NUMBERI9) 



9b. other REPORT NO(S) (Any other number* that may be am alined 
thia report) 



(0. DISTRIBUTION STATEMENT 



yffhis document has been approved for publf<J 

ise and sale; Its distribution is unlimited, 



)(. SUPPLEMENTARY NOTES 



12. SPONSORING MILI TARY ACTIVITY 



Naval Postgraduate School 
Monterey, California 93940 



13. ABSTRACT 



A study of the physics of the piano reveals that while the upper partial s 
of the steel strings are the eigen-frequencies of the complex tone, they are not 
integer multiples of the respective fundamentals. To properly measure and tune 
these eigen-partials, a digital filter capable of sweeping a major portion of the 
audio-frequency spectrum had to be implemented. Such a filter, a tuneable 
fixed-coefficient digital filter, is discussed as well as a simple pole-zero 
design procedure for determining the required coefficients. Each module, 
including the Frequency Deviation Detector and Counter, the Time-Base Generator, 
the Digital Filter, the Reference Frequency Generator and the Display and 
Control Module, of the proposed tuner is illustrated and discussed. 



DD 



FORM 



\AT\ (PAGE 1) 
i nov es I ™T # w 

S/N 0101-807-681 1 



93 



Security Classification 



A-31408 



Security Classification 



KEY WO RDS 



Digital Filtering 

Tuneable Digital Filtering 

Frequency Deviation Detector and Counter 

Equal -Tempered Keyboard Tuner 



DD 



FORM 



1473 < BACK 



S/N 0101-807-6821 



98 



Security Classification 



DUDLEY KNOX LIBRARY 



3 2768 00453668